Time evolution of the probability distribution of returns in the Heston model of stochastic...
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Time evolution of the probability distribution of returns in the Heston
model of stochastic volatility compared with the high-frequency
stock-market data
Victor M. YakovenkoA. Christian SilvaRichard E. Prange
Department of Physics University of Maryland College Park, MD, USA
APFA-4 Conference, Warsaw, Poland, 15 November 2003
Mean-square variation of log-return as a function of time lag
The log-return is xt = ln(S2/S1)-t, where S2 and S1 are stock prices at times t2 and t1, t = t2t1 is the time lag, and is the average growth rate.
1863: Jules Regnault in “Calcul des Chances et Philosophie de la Bourse” observed t
2 = xt2 t for the
French stock market.
See Murad Taqqu http://math.bu.edu/people/murad/articles.html 134 “Bachelier and his times”.
1900: Louis Bachelier wrote diffusion equation for the Brownian motion (1827) of stock price: Pt(x) exp(-x2/2vt) is Gaussian.
However, experimentally Pt(x) is not Gaussian, although xt2 = vt
Models with stochastic variance v: xt2 = vt = t.
1993: Steve Heston proposed a solvable model of multiplicative Brownian motion for xt with stochastic variance vt:
( );12tv
t t tdx dt v dW ( )( ) 2t t t tdv v dt v dW
Wt(1) & Wt
(2) are Wiener processes. The model has 3 parameters: - the average variance: t
2 = xt2 = t.
- relaxation rate of variance, 1/ is relaxation time - volatility of variance, use dimensionless parameter = 2/2
What is probability distribution Pt(x) of log-returns as a function of time lag t?
Solution of the Heston modelDragulescu and Yakovenko obtained a closed-form analytical formula for Pt(x) in the Heston model: cond-mat/0203046,Quantitative Finance 2, 443 (2002), APFA-3:
( )( ) ,2tF kikxdk
tP x e e
characteristic function ( )( ) tF k
tP k e
( ) ln cosh sinh ,2 1
2 2 2 2t t t
tF k ( ) ,2
01 kx 0x
where t t is the dimensionless time.
Short time: t « 1: exponential distribution For =1, it scales
( ) exp2
tP x xt
( ) /t tP x f x
Long time: t » 1: Gaussian distributionIt also scales
( ) exp2
2t
xP x
t
( ) /t tP x g x
Comparison with the data
Previous work: Comparison with stock-market indexes from 1 day to 1 year. Dragulescu and Yakovenko, Quantitative Finance 2, 443 (2002),cond-mat/0203046; Silva and Yakovenko, Physica A 324, 303 (2003), cond-mat/0211050.
New work: Comparison with high-frequency data for several individual companies from 5 min to 20 days. The plots are for Microsoft (MSFT).Silva, Prange, and Yakovenko (2003) = 3.8x10-4 1/day = 9.6 %/year, 1/=1:31 hour, =1
Cumulative probability distribution
For short time t ~ 30 min – several hours:exponential
Solid lines – fits to the solution of the Heston model
For very short time t ~ 5 min:Power-law (Student)
For long time t ~ few days: Gaussian
Short-time and long-time scaling
GaussianExponential
From short-time to long-time scaling
(t)
Characteristic function can be directly obtained from the data
( )
Re t
t
t
ikx
x
P k
e
Direct comparisonwith the explicitformula for theHeston model:
( )( ) tF ktP k e
Brazilian stock market indexFits to the Heston model by Renato Vicente and
Charles Mann de Toledo, Universidade de Sao Paulo
= 1.4x10-3 1/day = 35 %/year, 1/ = 10 days, = 1.9
Comparison with the Student distribution
1
2 2 2
1( )
1 /t d
t
P x
x
The Student distribution works for short t, but does not evolve into Gaussian for long t.
Conclusions• The Heston model with stochastic variance well describes
probability distribution of log-returns Pt(x) for individual stocks from 15 min. to 20 days.
• The Heston model and the data exhibit short-time scaling Pt(x)exp(2|x|/t) and long-time scaling Pt(x)exp(x2/2t
2). For all times, t
2 = xt2 = t.
• For individual companies, the relaxation time 1/ is of the order of hours, but, for market indexes, 1/ is of the order of ten days.
• The Heston model describes Brazilian stock market index from 1 min. to 150 days.
• The Student distribution describes Pt(x) for short t, but does not evolve into Gaussian for long t.